View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topbgy Vol. 26. No. 2. pp. X9-110. 1987 0040-9383’87 13.00+ 00 mntcd ,n Great Bnlaln. e 1987 Pcrgamon loumals Ltd GENUS IS SUPERADDITIVE UNDER BAND CONNECTED SUM DAVID GABAI* (Receiced 22 July 1986) DEFINITIONS. Let k, , k, be two knots in S’ which can be separated by an embedded 2-sphere. Let b: I x I + S3 be an embedding such tharfor i = 0,l f- ’ (ki) = I x i. k = k0 #bkl = (k, v k, u b(SZ x I)) - b(f x ?I) is called the band connected sum of k, and k,. b Is the trivial band and k = kO #k, is a connected sum if there exists a two-sphere Q separating k, from k, such that Q n b(Z x I) is a connected arc. If S,, and Si are disjoint Seifert surfaces for k, and k,, disjoint from b(l x f), then S = So u S, u b(l x I) is called the band connected sum of S,, and St. X is said to be obtained byfilling the 3-manifold M along the essential simple closed curve a c P, Pa toral component of dM if X is obtained by first attaching a two-handle to A4 along cz and then capping off the resulting two-sphere with a 3-cell. The purpose of this note is to prove the following result. THEOREM 1. If k is a band connected sum of k, and k,, then genus k 2 genus k, + ge?nts k,. Equality holds if and only if there exists a Seifert surface for k which is a band connected sum (using the same band) of minimal genus Seifert surfaces for k, and kz. Remarks. This gives a positive proof of the twenty year old Tristram-Lickorish conjecture (see also problem 1, [Z]), that genus is superadditive under band connected sum. In particular we obtain the main result of [3] “If the unknot is a band connected sum of k, and k,, then each ki is the unknot and the band is the trivial band.” In [4] Thompson gave a new proof of the above mentioned result of Scharlemann. This paper is a refinement of that proof, Theorem 1 has been independently proven, using different methods by Marty Scharlemann. Proof: Let D be an embedded disc in S3 such that D n (k u b(l x I)) = b(l x l/2) c b. Let L = ?D. Let S be an oriented surface of minimal genus in S3 -.6(L) such that ;S = k. We can assume that S n D is an arc. If L bounds a disc disjoint from S, then k is a connected sum of k, and k, and b is the trivial band. In this case after finding a splitting two-sphere Q and an S such that S n Q = S n D it is immediate that S is a minimal genus surface for k and is also a connected sum of minimal genus surfaces for k, and k,, so Theorem 1 follows. Now assume that k Ti k, #k,. Let M = S3 - fi (k u L) and P = SN(L). If there exists no non-boundary parallel torus T c M separating S from P, then A4 is S,-atoroidal (see Definition 1.6 Cl]). *Partially supported by National Science Foundation research grant DMS84-03645. 209 210 David Gabai If .\I is not S,-atoroidal, then S is a minimal genus surface fork in S3. To see this let T c M be a non-boundary parallel torus disjoint from S which separates P from S. Tcan be isotoped in M so that T n D is a boundary parallel simple closed curve. T is compressible in W, the closure of the component of S3 - T containing k, so Wis a solid torus. W is knotted in S3, else T would be boundary parallel in SI. Therefore, T is incompressible in S3 - @, so S is a surface of minimal genus for k in S3. We now show that if M is S,-atoroidal, then S is a minimal genus surface for k in S3. By Corollary 2.4 [l], with at most one exception (up to isotopy), the following holds. If N is obtained by filling M along an essential simple closed curve in P, then N is irreducible and genus S I genus S’ if S’ is any embedded oriented surface in N with dS’ = ZS. By Thompson [4] filling .M along the curve ZD yields a reducible manifold. [She observes that in S2 x S’, the manifold obtained by performing O-frame surgery to L, k can be isotoped to k, #k, by a series of applications of the “light bulb trick”. So M filled along dD equals S2 x S’ # (S3 - 1”5(k, #k,).] Therefore dD must correspond to the exceptional filling and S remains minimal genus when M is filled along the meridian of N(L). Let R be the surface S--&‘(D). Note that ZR = kO u k,. Let R’ be a surface obtained by compressing R a maximal number of times. Since k, and k, could be separated by a two sphere, R’ has two components, respectively spanning kO and k,. We obtain genus k = genus S = l/2(2 -z(R)) 2 l/2 (2 - z(R’)) 2 genus k, + genus k,. Therefore genus k = genus k,, + genus k, ifand only if R is a union of minimal genus surfaces for kO and k,, i.e. S is a band connected sum of minimal genus surfaces for kO and k,. REFERENCES 1. D. GABAI: Foliations and the Topology of 3-Manifolds II, J. of&ff: Geom. (to appear). 2. R. KIRLW:Problems in low dimensional manifold theory, Proc. Symp. Pure Mach. AMS 32 (1978). 273-312. 3. M. SCHARLEMANN:Smooth spheres in W’with four critical points are standard, Inumr. Marh. 79 (1985), 125-141. 4. A. THOMPSON:Property P for the bandconnect sum of two knots, Topology (1986), 205-207. University of Pennsylvania lnstitut des Hautes Etudes Scientijiques Current Address: California Institute of Technology Pasadena, CA 91125 .